Question

Radioisotopes of iodine are widely used in medicine. For example, iodine-131 \(\left(t_{1 / 2}=8.04\right.\) days) is used to treat thyroid cancer. If you ingest a sample of \(\mathrm{Na}^{131} \mathrm{I},\) calculate the time required for the isotope to decrease to \(5.0 \%\) of its original activity.

Short answer

The time required is approximately 35.8 days.

Step-by-step solution

Understanding Half-Life

The half-life of iodine-131 is 8.04 days. This means that every 8.04 days, half of the iodine-131 decays.

Recognizing the Exponential Decay Equation

The activity of a radioactive substance is given by the equation \( A = A_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \), where \( A \) is the remaining activity, \( A_0 \) is the initial activity, \( t \) is the time elapsed, and \( t_{1/2} \) is the half-life.

Setting Up the Equation for 5% Activity

To find when the activity is 5% of the original, set \( A = 0.05 A_0 \). Substitute into the equation: \( 0.05 A_0 = A_0 \times \left( \frac{1}{2} \right)^{\frac{t}{8.04}} \). Cancel \( A_0 \) from both sides to get \( 0.05 = \left( \frac{1}{2} \right)^{\frac{t}{8.04}} \).

Solving the Exponential Equation

Take the logarithm of both sides: \( \ln(0.05) = \ln\left( \left( \frac{1}{2} \right)^{\frac{t}{8.04}} \right) \). This simplifies to \( \ln(0.05) = \frac{t}{8.04} \times \ln\left( \frac{1}{2} \right) \).

Isolating \(t\)

Solve for \( t \) by rearranging the equation: \( t = \frac{8.04 \times \ln(0.05)}{\ln(\frac{1}{2})} \).

Calculating \(t\)

Use a calculator to compute \( t = \frac{8.04 \times \ln(0.05)}{\ln(0.5)} \). The calculated time \( t \) is approximately 35.8 days.

Key concepts

Half-life

The concept of half-life is key to understanding radioactive decay. It represents the time required for half of a radioactive substance to decay. For iodine-131, the half-life is 8.04 days. This means that every 8.04 days, the amount of iodine-131 decreases by half.
Half-life is a constant value specific to each radioactive material. It doesn't depend on how much substance you start with.

• If you start with 10 units, after one half-life, you'll have 5 units left.
• After two half-lives, you'll have 2.5 units left, and so on.

Understanding half-life helps predict how long it will take for a substance to become either safe or ineffective.

Iodine-131

Iodine-131 is a radioactive isotope of iodine. It's commonly used in medical applications, particularly in treating thyroid cancer. Due to its radioactive nature, iodine-131 emits particles that can destroy diseased thyroid tissues.
This isotope is carefully administered in controlled doses to target and treat specific areas in the thyroid gland.

• Its half-life of 8.04 days is significant enough to allow for effective treatment over a set period.
• Post-treatment, the remaining iodine-131 breaks down, reducing radiation exposure over time.

Working with iodine-131 requires understanding its half-life to ensure it is used effectively and safely.

Exponential Decay

Exponential decay is a process where the quantity decreases at a rate proportional to its current value. Radioactive substances like iodine-131 follow this pattern. The decay formula used to track remaining radioactive activity is:
\[ A = A_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]
Where:

• \( A \) is the remaining activity at time \( t \).
• \( A_0 \) is the initial activity.
• \( t \) is the elapsed time.
• \( t_{1/2} \) is the half-life.

This formula shows how each period (such as 8.04 days for iodine-131) reduces the activity by half. For the case where activity needs to decline to 5%, you set \( A = 0.05 A_0 \) and solve for \( t \). This calculation shows how long it takes for the substance to reduce to a specific amount of its original value. Understanding exponential decay is essential to predict the long-term behavior of radioactive materials.